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Properties

Label 2-78-3.2-c2-0-0
Degree $2$
Conductor $78$
Sign $-0.209 - 0.977i$
Analytic cond. $2.12534$
Root an. cond. $1.45785$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−2.93 + 0.628i)3-s − 2.00·4-s + 6.81i·5-s + (0.888 + 4.14i)6-s − 9.58·7-s + 2.82i·8-s + (8.21 − 3.68i)9-s + 9.64·10-s + 15.6i·11-s + (5.86 − 1.25i)12-s − 3.60·13-s + 13.5i·14-s + (−4.28 − 20.0i)15-s + 4.00·16-s − 15.6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.977 + 0.209i)3-s − 0.500·4-s + 1.36i·5-s + (0.148 + 0.691i)6-s − 1.36·7-s + 0.353i·8-s + (0.912 − 0.409i)9-s + 0.964·10-s + 1.41i·11-s + (0.488 − 0.104i)12-s − 0.277·13-s + 0.968i·14-s + (−0.285 − 1.33i)15-s + 0.250·16-s − 0.918i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ -0.209 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.295901 + 0.365958i\)
\(L(\frac12)\) \(\approx\) \(0.295901 + 0.365958i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (2.93 - 0.628i)T \)
13 \( 1 + 3.60T \)
good5 \( 1 - 6.81iT - 25T^{2} \)
7 \( 1 + 9.58T + 49T^{2} \)
11 \( 1 - 15.6iT - 121T^{2} \)
17 \( 1 + 15.6iT - 289T^{2} \)
19 \( 1 + 24.1T + 361T^{2} \)
23 \( 1 + 8.09iT - 529T^{2} \)
29 \( 1 - 33.5iT - 841T^{2} \)
31 \( 1 - 31.1T + 961T^{2} \)
37 \( 1 - 40.8T + 1.36e3T^{2} \)
41 \( 1 - 60.5iT - 1.68e3T^{2} \)
43 \( 1 - 18.6T + 1.84e3T^{2} \)
47 \( 1 - 6.55iT - 2.20e3T^{2} \)
53 \( 1 - 39.5iT - 2.80e3T^{2} \)
59 \( 1 + 48.4iT - 3.48e3T^{2} \)
61 \( 1 + 33.4T + 3.72e3T^{2} \)
67 \( 1 - 0.352T + 4.48e3T^{2} \)
71 \( 1 - 99.3iT - 5.04e3T^{2} \)
73 \( 1 + 94.9T + 5.32e3T^{2} \)
79 \( 1 - 51.4T + 6.24e3T^{2} \)
83 \( 1 - 87.9iT - 6.88e3T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + 82.7T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.57342242662614810221455893046, −13.01482397662881854344060548305, −12.28570226986816266763823569283, −11.10524438238504702178376565540, −10.17408851123103965156708792051, −9.592401233326812053816982059778, −7.12413502117573081865003675125, −6.34234441030474608742663951051, −4.46273320406835549037282275345, −2.80366897464399009301389161264, 0.43753817902222293959131046279, 4.19725031394490334049681779170, 5.72012264807523264129784738182, 6.40129905178835053584719907057, 8.116921444955758922170314436054, 9.226189580835246702671685931691, 10.48050195703427345802917218551, 12.03896768519640560257033413863, 12.95712422853158336676333259991, 13.47880364804902469450759158792

Graph of the $Z$-function along the critical line